Page:Popular Science Monthly Volume 58.djvu/269

Rh intersection of a line and circle which are co-planar and non-intersecting in the ordinary sense. It is impossible to exaggerate the importance of these conceptions. Without them the beautiful fabric of modern geometry would not stand a moment. It will be seen to many readers, no doubt, that a fabric built upon such a foundation will have very much the same stability as a 'castle in Spain.' Such, however, is far from the case. The analysis by which our operations proceed is a thoroughly well founded and trustworthy instrument, and when we give to it the geometric interpretation which we are entirely justified in doing, we find frequently that it reveals to us facts which our senses unaided by its finer powers of interpretation could not have discovered. These facts require for their adequate explanation the recognition of the so-called imaginary elements of the figure. Let us take one more illustration. If from a point outside of, but in the same plane with, a circle we draw two tangents to the circle and connect the points of tangency with a straight line, the original point and the line last mentioned stand in an important relation to each other and are called respectively pole and polar with regard to the circle. Now suppose the point is inside the circle. The whole construction just described becomes then geometrically impossible, but analytically we can draw from a point within a circle two imaginary tangents to the circle, and similarly we can connect the imaginary points of tangency by a straight line, and this straight line is found to be a real line. Moreover, in its relations to the point and circle it exhibits precisely the same properties which are found in the case of the pole and polar first described. Hence this point and line are also included in the general definition of pole and polar. Such examples might be multiplied indefinitely, but they would all go to emphasize the fact of the great power of generalization which resides in the methods of analytic geometry.

While the power of the analytic method as an instrument of research is far greater than that of the older pure geometric method, yet to many minds it lacks somewhat the beauty and elegance of that method as an intellectual exercise. This is due to the fact that its operations, like all algebraic operations, are largely mechanical. Given the equations representing a certain geometric condition, we subject these equations to definite transformations and the results obtained denote certain new geometric conditions. We have been whisked from the data to the result very much as we are hurried over the country in a railroad train. We may have noted the features of the country as we passed through it or we may not; we arrive at our destination just the same. Pure geometric research, on the other hand, resembles travel on foot or horseback. We must scrutinize the landmarks and keep a careful watch on the direction in which we are traveling, lest we take •-a wrong turn and fail to reach our destination. The result is that we